Related papers: The equation w(x,y)=u over free groups

We construct a Fock space associated to a symmetric function $Q:U\times U \to (-1,1)$, where $U$ is a nonempty open subset of $\mathbb R^j$ for some $j$. Namely, we will have operator-valued distributions $a(x)$ and $a^+(y)$ satisfying…

In one of our recent papers, the associative and the Lie algebras of Weyl type $A[D]=A\otimes F[D]$ were defined and studied, where $A$ is a commutative associative algebra with an identity element over a field $F$ of any characteristic,…

Consider a group word w in n letters. For a compact group G, w induces a map G^n \rightarrow G$ and thus a pushforward measure {\mu}_w on G from the Haar measure on G^n. We associate to each word w a 2-dimensional cell complex X(w) and…

Sets of solutions to finite systems of equations in a free group, are equivalent to sets of homomorphisms from a fixed f.p. group into a free group. The latter can be encoded in a diagram, the construction of which is valid also for f.g.…

W.~Magnus' representations of submonoids $ E \leq \mbox{End}(F) $ of the endomorphisms of a free group $ F $ of finite rank are generalised by identifying them with the first homology group of $ F $ with particular coefficient modules. By…

A generic polynomial f(x,y,z) with a prescribed Newton polytope defines a symmetric spatial curve f(x,y,z)=f(y,x,z)=0. We study its geometry: the number, degree and genus of its irreducible components, the number and type of singularities,…

Ostrowski's theorem implies that $\log(x),\log(x+1),\ldots$ are algebraically independent over $\mathbb{C}(x)$. More generally, for a linear differential or difference equation, it is an important problem to find all algebraic dependencies…

Let $K[x,y]$ be the polynomial algebra in two variables over an algebraically closed field $K$. We generalize to the case of any characteristic the result of Furter that over a field of characteristic zero the set of automorphisms $(f,g)$…

For any integers $m,n$ with $m\ne 0$ and $n>0$, let $G_{m,n}$ denote the group presented by $\langle x,y,z\mid x=[z^m,x][z^n,y]\rangle$; for any integers $m,n>0$, let $H_{m,n}$ denote the group presented by $\langle x,y,z\mid…

We study the existence of homomorphisms between Out(F_n) and Out(F_m) for n > 5 and m < n(n-1)/2, and conclude that if m is not equal to n then each such homomorphism factors through the finite group of order 2. In particular this provides…

Let $[n]^{(k)}$ be the set of all ordered $k$-tuples of distinct elements in $[n]=\{1,2,...,n\}$. The $(n,k,r)$-arrangement graph $A(n,k,r)$ with $1\leq r\leq k\leq n$, is the graph with vertex set $[n]^{(k)}$ and with two $k$-tuples are…

If G is a (connected) complex Lie Group and Z is a generalized flag manifold for G, the the open orbits D of a (connected) real form G_0 of G form an interesting class of complex homogeneous spaces, which play an important role in the…

In this short note, we show that the generalized type semigroup $\CW(X, \Gamma)$ introduced by the author in \cite{M3} belongs to the category \textnormal{W}. In particular, we demonstrate that $\CW(X, \Gamma)$ satisfies axioms (W1)-(W4)…

Given a Tychonoff space $X$, let $F(X)$ and $A(X)$ be respectively the free topological group and the free Abelian topological group over $X$ in the sense of Markov. In this paper, we provide some topological properties of $X$ whenever one…

We generalize the combinatorial approaches of Rapaport and Higgins--Lyndon to the Whitehead algorithm. We show that for every automorphism $\varphi$ of a free group $F$ and every word $u\in F$ there exists a finite multiset of words…

We show that the freeness problems for automaton semigroups and for automaton monoids are undecidable and, thereby, solve an open problem listed by Grigorchuk, Nekrashevych and Sush\-chansk\u{\i}i. We achieve this using a new technique to…

We give conditions of an extension of a free group to be hyperbolic and relatively hyperbolic using the dynamics of the action of $\out$ on the complex of free factors combined with the weak attraction theory. We work with subgroups of…

Automorphisms of the quantum Schubert cell algebras ${\mathcal U}_q^\pm[w]$ of De Concini, Kac, Procesi and Lusztig and their restrictions to some key invariant subalgebras are studied. We develop some general rigidity results and apply…

The Whitehead Model of free groups can be used to measure the complexity, or degree, of automorphisms of free groups. The bound for the degree of the $f \circ g$ for deg$(f) =$ deg($g) = 0$ had previously been discovered. We extend this…

Let \sigma be an automorphism of a commutative k-algebra R. The skew polynomial ring R[t;\sigma] is generated by R and an indeterminate t subject to the relations ta=\sigma(a)t for all a in R. For certain R and appropriate \sigma there are…